Euler-Lagrange form

principle of extremal action, Euler-Lagrange equations, de Donder-Weyl formalism

Given a smooth bundle $E \to \Sigma$ with $\Sigma$ a smooth manifold of dimension $n$, then a horizontal $n$-form $\mathbf{L}$ on the jet bundle $Jet(E)$ is a local Lagrangian. Its de Rham differential has a unique decomposition into a source form $\mathbf{E}$ and a horizontally exact form (with respect to the variational bicomplex)

$d \mathbf{L} = \mathbf{E} - d_H \theta
\,.$

This source form $\mathbf{E}$ is the *Euler-Lagrange form* of $\mathbf{L}$. It vanishes precisely at those points which are solutions to the Euler-Lagrange equations induced by $\mathbf{L}$.

The combination $\rho \coloneqq \mathbf{L} + \theta$ is the corresponding Lepage form.

- G. J. Zuckerman,
*Action principles and global geometry*, in Mathematical Aspects of String Theory, S. T. Yau (Ed.), World Scientific, Singapore, 1987, pp. 259€284. (pdf)

Last revised on November 8, 2017 at 15:17:59. See the history of this page for a list of all contributions to it.